STOR 435: Introduction to Probability, Spring 2008
Study Guide for the Final Exam
A. Study Guide for Midterm 1
1. Basics of counting: combinations and permutations, the binomial
theorem.
2. Basic rules of Boolean algebra.
3. Axioms of probability and their immediate consequences (e.g.
inclusion-exclusion).
4. Sample spaces with equally likely outcomes.
5. Conditional probability: basic definition and properties.
6. Disjoint events and partitions.
7. The multiplication rule, law of total probability and Bayes' formula.
8. Conditional probabilities are probabilities when the conditioning
event is fixed.
9. Definition and basic properties of independence. Disjoint is different from independent.
10. Random variables: definition, ways of finding probabilities,
``when''
11. Discrete random variables: possible values and probability mass
function.
12. Expectation of a discrete random variable: definition and basic
properties
13. How to find the distribution of a function of a random variable.
14. How to find the expectation of a function of a random variable.
15. Variances: definition and basic properties.
16. Bernoulli trials
17. Binomial, geometric and negative binomial random variables:
definition, pmf, expectation, variance
18. Poisson random variable: pmf, interpretation, application.
B. Study Guide for Midterm 2
1. Basic definition of a continuous random variable, probability
density function
2. Relationship between pdf and CDF
3. Expectations and Variance, basic properties
4. Expectations of functions of a r.v., and formula for expectation in
terms of P(X > x)
5. Basic continuous random variables: densities, CDFs, expectations,
variances
a. Uniform
b. Normal (finding normal probabilities using
tables)
c. Exponential (memoryless property)
d. Gamma family of distributions, and the Gamma
function
e. Cauchy
6. The CDF method
7. The normal approximation to the binomial
8. Jointly distributed random variables
9. Joint pmf and pdf. Relationship between joint and marginal
distributions
10. Finding probabilities involving two jointly distributed continuous
r.v.
11. Basic multivariate calculus
12. Independent random variables: definition, connections with joint
pmf and joint pdf
13. Sums of independent random variables: Poisson, Gamma, Normal,
Binomial
14. Conditional distributions: Definitions and properties of
conditional pmf, pdf
C. Additional Material
1. Definition of
Eg(X,Y) in discrete and continuous cases
2. Basic properties of
expectations: Monotonicity and Linearity
3. Method of indicator functions, inc. basic properties of indicator
random variables
4. Sample mean of n random variables: definition, mean and variance
5. Independence and the expectation of products
6. Covariance: definition and basic properties
7. Sample variance: definition
8. Moment generating functions: definition and basic properties
9. Markov, Chebyshev and One-sided Chebyshev inequalities
10. Weak law of large numbers
11. Central limit theorem: ``A standardized sum of n i.i.d. random
variables is approximately normal when n is large''.
12. Jensen's inequality
13. Cauchy-Schwartz inequality